Isomorphism between $\mathbb Q$ under addition and $\mathbb Q^+$ under multiplication?

An exercise asks me to show that $\mathbb Q \times C_2$ (under addition of the rationals) is isomorphic to $\mathbb Q-\{0\}$ under multiplication. Quotienting $\mathbb Q \times C_2$ by $C_2$ gives $\mathbb Q$ under addition. The only element of order 2 in $\mathbb Q-\{0\}$ is -1, so quotienting out by $<-1>$ gives $\mathbb Q^+$ under multiplication. But I don't think $\mathbb Q$ under addition and $\mathbb Q^+$ under multiplication can be isomorphic. If $f$ were such an isomorphism from the additive group to the multiplicative group, then $f\left(\frac{a}{b}\right)$ would have to be $f(1)^\frac{a}{b}$. But no matter which positive rational $r$ I pick to be $f(1)$ there will be a rational $q$ such that $r^q$ is irrational, unless I let $r=1$ in which case $f$ would not be injective.

Where did I mess up? Am I not allowed to say that the quotients of both groups by $C_2$ are isomorphic?

• what's $C_2$ ? (obligatory extra characters) – Stefan Smith Dec 31 '13 at 1:10
• The cyclic group of order $2$, I assume. – Dylan Yott Dec 31 '13 at 1:10
• – Dylan Yott Dec 31 '13 at 1:11
• Yes, it's the cyclic group of order 2. So did the exercise ask me to prove a false statement? – Nishant Dec 31 '13 at 1:19
• @Nishant Yes, the exercise is wrong. – Andreas Blass Dec 31 '13 at 1:24